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# M32-13

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Math Expert
Joined: 02 Sep 2009
Posts: 49241

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17 Jul 2017, 10:20
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Difficulty:

75% (hard)

Question Stats:

27% (00:55) correct 73% (01:03) wrong based on 37 sessions

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If $$n = x^5*y^7$$, where $$x$$ and $$y$$ are positive integers greater than 1, then how many positive divisors does $$n$$ have?

(1) $$x$$ does not have a factor $$p$$ such that $$1 < p < x$$ and $$y$$ does not have a factor $$q$$ such that $$1 < q < y$$.

(2) $$n$$ has only two prime factors.

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17 Jul 2017, 10:20
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Official Solution:

If $$n = x^5*y^7$$, where $$x$$ and $$y$$ are positive integers greater than 1, then how many positive divisors does $$n$$ have?

(1) $$x$$ does not have a factor $$p$$ such that $$1 < p < x$$ and $$y$$ does not have a factor $$q$$ such that $$1 < q < y$$. This statement implies that both $$x$$ and $$y$$ are primes (a prime number does not have a factor which is greater than 1 and less than itself, it has only two factors 1 and itself). Now, if $$x$$ and $$y$$ are different primes, then the number of factors of $$n$$ will be $$(5+1)(7+1)=48$$ but if $$x=y$$, then $$n = x^5*y^7=x^{12}$$ and it will have 13 factors. Not sufficient.

(2) $$n$$ has only two prime factors. If those primes are $$x$$ and $$y$$, then the number of factors of $$n$$ will be $$(5+1)(7+1)=48$$ but if, say $$x=2*3=6$$ and $$y=2$$, then $$n = x^5*y^7=2^{12}*3^5$$ and it will have $$(12+1)(5+1)=78$$ factors. Not sufficient.

(1)+(2) Since from (2) $$n$$ has only two prime factors, then from (1) it follows that $$x$$ and $$y$$ are different primes so the number of factors of $$n$$ will be $$(5+1)(7+1)=48$$. Sufficient.

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19 Jul 2017, 00:20
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What about when x is exactly 1? That satisfies statement one (x has no factor greater than 1 and less than itself), but would lead to answer E eventually.

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19 Jul 2017, 02:16
spence11 wrote:
What about when x is exactly 1? That satisfies statement one (x has no factor greater than 1 and less than itself), but would lead to answer E eventually.

Posted from my mobile device

Thank you. Edited to rule out this case.
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23 Feb 2018, 18:02
Statement 1 raises the fact that x and y could be different or the same. statement 2 says that n has two prime factors. when the statements are combined together it is possible for x is equal to 6 and y is 2. Here the n has two prime factors and they are different. Also x could equal 2 and y equal 3. Both scenarios give us different factors. The answer should be E.
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24 Feb 2018, 00:31
etibamanya wrote:
Statement 1 raises the fact that x and y could be different or the same. statement 2 says that n has two prime factors. when the statements are combined together it is possible for x is equal to 6 and y is 2. Here the n has two prime factors and they are different. Also x could equal 2 and y equal 3. Both scenarios give us different factors. The answer should be E.

The answer should be and is C, not E.

(1) means that x and y are primes, so x cannot be 6. 6 does have factors which are more than 1 and less than 6 itself: 2 and 3.
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24 Feb 2018, 08:11
Is there a place where these questions are just listed? I notice that often the title of the post is something like "Mxy-zw" - are they being taken from a book or something? And how do I get my hands on it?

Thanks.
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01 Sep 2018, 09:42
My initial instinct in seeing two variables x and y is that they are different values. Based on this answer explanation I should completely scrub this line of thinking from my mind for the GMAT, right?
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02 Sep 2018, 03:02
pzone123 wrote:
My initial instinct in seeing two variables x and y is that they are different values. Based on this answer explanation I should completely scrub this line of thinking from my mind for the GMAT, right?

Right. Unless it is explicitly stated otherwise, different variables CAN represent the same number.
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Re: M32-13 &nbs [#permalink] 02 Sep 2018, 03:02
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# M32-13

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